If the first term of a harmonic progression is 5 and the second term is 10, what

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Question: If the first term of a harmonic progression is 5 and the second term is 10, what is the third term?

Options:

Correct Answer: 25

Solution:

The reciprocals are 1/5 and 1/10. The common difference is -1/10. The third term\'s reciprocal will be 1/10 - 1/10 = 1/15, so the third term is 15.

If the first term of a harmonic progression is 5 and the second term is 10, what is the third term?

If the first term of a harmonic progression is 5 and the second term is 10, what is the third term?

Step 1: Identify the first term of the harmonic progression (HP), which is 5.
Step 2: Identify the second term of the harmonic progression (HP), which is 10.
Step 3: Find the reciprocals of the first and second terms. The reciprocal of 5 is 1/5, and the reciprocal of 10 is 1/10.
Step 4: Calculate the common difference between the reciprocals. Subtract 1/5 from 1/10.
Step 5: To subtract, convert 1/5 to a fraction with a common denominator of 10. This gives us 2/10.
Step 6: Now subtract: 1/10 - 2/10 = -1/10. This is the common difference.
Step 7: To find the third term's reciprocal, take the second term's reciprocal (1/10) and subtract the common difference (-1/10).
Step 8: So, 1/10 - (-1/10) = 1/10 + 1/10 = 2/10 = 1/5.
Step 9: The reciprocal of the third term is 1/15, so the third term is the reciprocal of 1/15, which is 15.

Harmonic Progression – A sequence of numbers is in harmonic progression if the reciprocals of the terms form an arithmetic progression.
Reciprocal Relationships – Understanding how to manipulate and calculate the reciprocals of terms in a harmonic progression.
Common Difference – The difference between consecutive terms in the sequence of reciprocals, which is crucial for finding subsequent terms.